Question

A thermally insulated piece of metal is heated under atmospheric pressure by an electric current so that it receives electric energy at constant power $P$ . This leads to an increase in the absolute temperature $T$ of the metal with time $t$ as $T\left(t\right)=T_0{\left(\left[1+a\left(t-t_0\right)\right]\right)}^{1/4}$ , where $a$ , $t_0$ and $T_0$ are constants. The heat capacity $C_P\left(T\right)$ of the metal is

• A. $\frac{4P}{a{T}_0}$
• B. $\frac{4P{T}^3}{a{T}_0^4}$
• C. $\frac{2P{T}^3}{a{T}_0^4}$
• D. $\frac{2P}{a{T}_0}$

Answer 1

Answer: (B)

Heat given to the metal $dQ=Pdt=C_P(t)dT$
At constant pressure in time interval at
Given, $T=T_0{\left[1+a\left(t-t_0\right)\right]}^{1/4}$
$\frac{dT}{dt}=\frac{T_0}{4}{\left[1+a\left(t-t_0\right)\right]}^{-3/4}\times a$
From Eqs. (i) and (ii) $C_P(T)=\frac{P}{\left(\frac{dT}{dt}\right)}=\frac{4P{\left[1+a\left(t-t_0\right)\right]}^{3/4}}{T_{0}a}=\frac{4P{T}^3}{a{T}_0^4}$